Pierre Berthelot

Crystalline cohomology and $\cal D$-module theoryw

Abstract: The classical equivalence between modules with integrable connection (or left ${\cal D}_X$-modules) and crystals, and the corresponding comparison theorem between de Rham and crystalline cohomologies, have been the starting point of the development of crystalline cohomology. On the other hand, the developpement of "algebraic analysis" (over $\bf C$) has lead to a full formalism of cohomological operations for ${\cal D}_X$-modules. However, it is well known that these operations are not compatible in an obvious way with the natural operations arising in crystalline cohomology. The purpose of this talk is to explain how the duality theory for (quasi)-coherent ${\cal O}_X$-modules can be used to relate these operations. Similar methods can also be used to compare the corresponding constructions of algebraic de Rham cohomology with compact supports in the classical context.